We did a family vacation in D.C. the summer before last (early August). Since several have answered you point-by-point already, I’ll give our general experiences and it’ll cover most of your points I think.
We stayed in Arlington, at a hotel within site of the Iwo Jima memorial. I don’t remember the exact name of it, but my wife probably does. PM me if you want any more specific info. It turned out to be a perfect area to stay-- the hotel was well under $100, I think around $70 a night. Not super fancy but it was clean and had a decently nice pool (important for our 7 and 10 year old boys). The neighborhood felt very safe and was a little less crazy busy than D.C. proper.
From there it was an easy few block walk to the Rosslyn Metro station. As mentioned, the metro is great for getting around. DO NOT RENT A CAR! The traffic in D.C. is nuts. Between the metro and a fair amount of hoofing it, we covered a lot of ground, even with two kids.
You buy a metro card for each person from a machine and put a certain amount of money on it. The amount defaults to $20- you can change it to less, but we thought that was a minimum and put $20 on each card. Even after going all over D.C. on the metro for a week, we still had a few bucks left on the cards when we were done.
As mentioned, the Smithsonian museums are all within walking distance of each other along the mall, so it’s easy to hit more than one a day. I think we averaged two a day-- I like to read all the info and linger, so I could have stayed longer, but with two kids we move through a museum pretty quickly. You could do 3 a day if you really wanted to cover a lot of ground. It’s more about how deep you want to get than logistics.
In my opinion the Air and Space Museum is a must-see, so hopefully you can convince your wife. I remember Archie Bunker’s chair and I think we saw the Fonz’s jacket-- if you go to the American History museum (and they haven’t been rotated out or something) they’re on prominent display and you can’t miss them.
We went to the National archives to see the Declaration of Independence, Constitution and Bill of Rights. I thought it was one of the highlights of the trip; the kids, not so much, unsurprisingly. The line to see them was really long- they only let small groups at a time into this dim, climate-controlled area where they’re displayed. The Declaration is so faded it’s almost gone-- apparently it spent years on display in a patent office opposite a window where the sun beat on it every day. Amazing how casually our national relics were treated back in the day-- the flag from the star-spangled banner has swatches cut out of it that were given to prominent people as souvenirs in the past :smack:
We walked all along the mall and saw most of the memorials. One day we did pay for a cushy air-conditioned tour bus to see some of the most prominent memorials-- WWII, the Vietnam Wall (very impressive), Korean War, Lincoln Memorial (soon after they had cleaned off the green paint some nutcase had splashed on it-- fortunately it was open, but unfortunately, there was a big scaffold next to it that made it hard to get good photos). It was nice to give the feet a rest that one day but we did plenty of walking along the mall and seeing some of the memorials twice. I would have liked to see the Jefferson Memorial, but it’s aways off from the mall and we were walked out that day.
We saw the Arlington Cemetery, and that was very moving, seeing the sheer volume of rows and rows of identical white crosses. The Kennedy eternal flame memorial looked surprisingly shabby though-- the original flame had malfunctioned at some point (not so eternal after all) and they had a not-impressive looking secondary flame going.
One place that I don’t think many tourists go to, but I would recommend, is the Pentagon. What an enormous structure close up! We went to see the 9-11 memorial, which I thought was well-done and very emotionally resonant. It’s a series of metal wave-shapes that come out of the ground, each with a small pool of water underneath. The waves point toward the Pentagon in, I assume, the direction the plane hit. Engraved on each one were a few names of the people who died in the crash. The Pentagon itself was kind of eerie-- it was on a Sunday so it was mostly deserted, but there were signs everywhere to not take pictures and to report anything suspicious. Of the few tourists who were there, I saw one couple taking pictures of each other with the Pentagon in the background, and security immediately materialized to tell them to stop (they didn’t get taken away though). I took a few shots of the memorial itself and my 10 year old nervously told me to stop so I didn’t get arrested. I told him it was fine as long as I didn’t point it at the Pentagon itself (not sure if that was actually true 
Another place to go I would recommend that many tourists don’t get to is the island that the Teddy Roosevelt Memorial is on (called Teddy Roosevelt Island I think, believe it or not). It was a pretty ambitious walk from our hotel, but not too bad, even with our 7 and 10 year olds (we do a lot of walking though). It’s a really pretty walk across a bridge over the Potomac, and the island itself is a wooded nature-preserve type area with a number of hiking trails. It felt like a really appropriate area for Teddy’s memorial, with hs love of nature and his establishing the National Park system. In the middle of the island is the Memorial, with a giant statue of Teddy holding his arms up (took a shot of each kid posing with their arms up in the same position). There are also several giant stone slabs with some of his more memorable quotes engraved.
It turned out to be a great trip, and the kids even seemed to have a great time, even though much of it you wouldn’t think of as kid-friendly.
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